I find it a little strange that you had trouble finding a formal proof. For instance, every calculus textbook I have ever seen has a proof, as do many elementary analysis textbooks. Also see en.wikipedia.org/wiki/Nth_term_test.
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Pete L. ClarkSep 13 '10 at 17:34

If we know that the sequence converges and merely wish to show it converges to zero, then a proof by contradiction gives a little more intuition here (although the direct proofs are simple and beautiful). Assume $a_n\to a$ with $a>0$, then for all $n>N$ for some large enough $N$ we have $a_n > a/2$ (take $\varepsilon = a/2$ in the definition of the limit). Now the sum diverges: $\sum_{n>N}a_n > \sum_{n>N}a/2 = \infty$. A similar argument works when $a<0$.